On the adjoint quotient of Chevalley groups over arbitrary base schemes

TL;DR

This paper investigates the structure of the adjoint quotient of split semisimple Chevalley group schemes over arbitrary base schemes, revealing conditions under which the Chevalley morphism is an isomorphism or only dominant, with explicit computations.

Contribution

It provides a detailed analysis of the adjoint quotient for Chevalley groups over arbitrary bases, including explicit descriptions and conditions affecting the morphism’s properties.

Findings

Chevalley morphism t/W -> g/G is an isomorphism except for Sp_{2n} over bases with 2-torsion.

Explicit computation of the adjoint quotient in classical cases.

Examples where the formation of the quotient commutes or does not commute with base change.

Abstract

For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W, we show that the Chevalley morphism t/W -> g/G is an isomorphism except for the group Sp_{2n} over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g -> g/G commutes, or does not commute, with base change on S.